A prime number is called *-elite* if only finitely many generalized Fermat
numbers
are quadratic residues modulo . Let be a prime. Write
with and odd. Define the length of the *b-Fermat period of * to be the minimal natural
number such that
Recently Müller and Reinhart derived three
conjectures on *-elite* primes, two of them being the following. (1) For every natural number there
is a *-elite* prime. (2) There are generalized elite primes with elite periods of arbitrarily large lengths. We
extend Müller and Reinhart's observations and computational results to further support above two conjectures. We
show that Conjecture 1 is true for and that for every possible length
there actually exists a generalized elite prime with elite period length .

Jeffrey Shallit 2009-06-20